Bulletin of the Iranian Mathematical Society Vol. 35 No. 2 (2009 ), pp 217-227.
HYERS-ULAM STABILITY OF FIBONACCI
FUNCTIONAL EQUATION
S.-M. JUNG
Communicated by Mohammad Sal Moslehian
Abstract. We solve the Fibonacci functional equation, f (x) =
f (x − 1) + f (x − 2), and prove its Hyers-Ulam stability in the class
of functions f : R → X, where X is a real Banach space.
1. Introduction
In 1940, Ulam gave a wide ranging talk before the Mathematics Club
of the University of Wisconsin in which he discussed a number of important unsolved problems (see [19]). Among those was the question
concerning the stability of homomorphisms:
Let G1 be a group and let G2 be a metric group with
a metric d(·, ·). Given any ε > 0, does there exist a
δ > 0 such that if a function h : G1 → G2 satisfies
the inequality d(h(xy), h(x)h(y)) < δ, for all x, y ∈ G1 ,
then there exists a homomorphism H : G1 → G2 with
d(h(x), H(x)) < ε, for all x ∈ G1 ?
In the following year, Hyers [8] affirmatively answered the question
of Ulam for the case where G1 and G2 are Banach spaces. Taking this
fact into account, the additive Cauchy functional equation f (x + y) =
f (x) + f (y) is said to satisfy the Hyers-Ulam stability. This terminology
MSC(2000): Primary: 39B82; Secondary: 39B52, 39B72.
Keywords: Fibonacci functional equation, Fibonacci number, Hyers-Ulam stability.
Received: 20 October 2008, Accepted: 22 December 2008.
c 2009 Iranian Mathematical Society.
217
218
Jung
is also applied to the case of other functional equations. Later, the result
of Hyers was generalized by Rassias (see [15]). It should be remarked
that Aoki [1] proved a particular case of Rassias’ theorem regarding
the Hyers-Ulam stability of additive functions earlier than Rassias (see
[14]). We can find in the books [3, 9, 12] a lot of references concerning the
stability of functional equations (see also [2, 4, 5, 6, 7, 10, 11, 16, 17, 18]).
Throughout this paper, we denote by Fn the nth Fibonacci number,
for n ∈ N. In particular, we define F0 := 0. It is well known that
Fn = Fn−1 + Fn−2 for all n ≥ 2. From this famous formula, we may
derive a functional equation
(1.1)
f (x) = f (x − 1) + f (x − 2),
which may be called the Fibonacci functional equation. A function
f : R → X will be called a Fibonacci function if it satisfies (1.1), for all
x ∈ R, where X is a real vector space.
By α and β we denote the positive root (respectively the negative
root) of the quadratic equation x2 − x − 1 = 0; i.e.,
√
√
1+ 5
1− 5
α=
and β =
.
2
2
For any x ∈ R, [x] stands for the largest integer that does not exceed x.
Here, we will solve the Fibonacci functional equation (1.1) and prove
its Hyers-Ulam stability for the class of functions f : R → X.
2. General solution of Fibonacci equation
Here, let X be a real vector space. We investigate the general solution of the Fibonacci functional equation (1.1). As we shall see in the
following theorem, the general solution of Fibonacci functional equation
is strongly related to the Fibonacci numbers Fn .
Theorem 2.1. Let X be a real vector space. A function f : R → X is a
Fibonacci function if and only if there exists a function g : [−1, 1) → X
such that
(2.1)
(
F[x]+1 g(x − [x]) + F[x] g(x − [x] − 1)
(x ≥ 0),
f (x) =
(−1)[x] F−[x]−1 g(x − [x]) − F−[x] g(x − [x] − 1) (x < 0).
Hyers-Ulam stability of Fibonacci functional equation
219
Proof. Since α + β = 1 and αβ = −1, it follows from (1.1) that
(
f (x) − αf (x − 1) = β [f (x − 1) − αf (x − 2)] ,
(2.2)
f (x) − βf (x − 1) = α [f (x − 1) − βf (x − 2)] .
By mathematical induction, we can easily verify that
(
f (x) − αf (x − 1) = β n [f (x − n) − αf (x − n − 1)] ,
(2.3)
f (x) − βf (x − 1) = αn [f (x − n) − βf (x − n − 1)] ,
for all x ∈ R and n ∈ {0, 1, 2, . . .}. If we substitute x + n (n ≥ 0) for
x in (2.3) and divide the resulting equations by β n (resp. αn ), and if
we then substitute −m for n in the resulting equations, then we obtain
the equations in (2.3) with m in place of n, where m ∈ {0, −1, −2, . . .}.
Therefore, the equations in (2.3) are true for all x ∈ R and n ∈ Z.
We multiply the first and the second equations of (2.3) by β and α,
respectively. If we subtract the first resulting equation from the second
one, then we obtain:
αn+1 − β n+1
αn − β n
f (x − n) +
f (x − n − 1),
α−β
α−β
for any x ∈ R and n ∈ Z.
For any given x ≥ 0, if we put n = [x] in (2.4), then it follows from
Binet’s formula (see [13, Theorem 5.6]) that
(2.4)
f (x) =
f (x) = F[x]+1 f (x − [x]) + F[x] f (x − [x] − 1).
If x < 0, then we put n = [x] = −|[x]| in (2.4). Since αβ = −1, by
Binet’s formula, we have,
f (x) =
=
α−|[x]|+1 − β −|[x]|+1
α−|[x]| − β −|[x]|
f (x − [x]) +
f (x − [x] − 1)
α−β
α−β
−1
α|[x]|−1 − β |[x]|−1
f (x − [x])
α−β
(αβ)|[x]|−1
+
−1 α|[x]| − β |[x]|
f (x − [x] − 1)
α−β
(αβ)|[x]|
= (−1)[x] F|[x]|−1 f (x − [x]) + (−1)[x]+1 F|[x]| f (x − [x] − 1)
= (−1)[x] F−[x]−1 f (x − [x]) − F−[x] f (x − [x] − 1) .
Since 0 ≤ x − [x] < 1 and −1 ≤ x − [x] − 1 < 0, if we define a function
g : [−1, 1) → X by g := f |[−1,1) , then we can see that f is a function of
the form (2.1).
220
Jung
Now, we assume that f is a function of the form (2.1), where g :
[−1, 1) → X is an arbitrary function. We show that f is a Fibonacci
function.
First, we assume that x ≥ 2. Since x > x − 1 > x − 2 ≥ 0 and
(x − 1) − [x − 1] = x − [x], it follows from (2.1) that

f (x) = F[x]+1 g(x − [x]) + F[x] g(x − [x] − 1),



f (x − 1) = F[x] g(x − [x]) + F[x]−1 g(x − [x] − 1),



f (x − 2) = F[x]−1 g(x − [x]) + F[x]−2 g(x − [x] − 1).
Thus, we get
f (x − 1) + f (x − 2)
= F[x] + F[x]−1 g(x − [x]) + F[x]−1 + F[x]−2 g(x − [x] − 1)
= F[x]+1 g(x − [x]) + F[x] g(x − [x] − 1)
= f (x).
Now, suppose that 1 ≤ x < 2. For this case, we have 0 ≤ x − 1 < 1
and −1 ≤ x − 2 < 0. Since [x] = 1, [x − 1] = 0 and [x − 2] = −1, it
follows from (2.1) that

f (x) = F2 g(x − [x]) + F1 g(x − [x] − 1),



f (x − 1) = F1 g(x − [x]) + F0 g(x − [x] − 1),



f (x − 2) = − [F0 g(x − [x]) − F1 g(x − [x] − 1)] .
Consequently, since F0 = 0 and F1 = F2 = 1, we obtain:
f (x − 1) + f (x − 2) = F2 g(x − [x]) + F1 g(x − [x] − 1) = f (x).
If 0 ≤ x < 1, then −1 ≤ x − 1 < 0 and −2 ≤ x − 2 < −1. In this case,
we have [x] = 0, [x − 1] = −1 and [x − 2] = −2. Hence, (2.1) implies
that

f (x) = F1 g(x − [x]),



f (x − 1) = F1 g(x − [x] − 1),



f (x − 2) = F1 g(x − [x]) − F2 g(x − [x] − 1).
Therefore, we have
f (x − 1) + f (x − 2) = F1 g(x − [x]) = f (x).
Hyers-Ulam stability of Fibonacci functional equation
221
Finally, assume that x < 0. In view of (2.1), we see that

f (x) = (−1)[x] F−[x]−1 g(x − [x]) − F−[x] g(x − [x] − 1) ,




f (x − 1) = (−1)[x]−1 F−[x] g(x − [x]) − F−[x]+1 g(x − [x] − 1) ,




f (x − 2) = (−1)[x]−2 F−[x]+1 g(x − [x]) − F−[x]+2 g(x − [x] − 1) .
These equations yield:
f (x − 1) + f (x − 2)
= (−1)[x] F−[x]+1 − F−[x] g(x − [x])
− F−[x]+2 − F−[x]+1 g(x − [x] − 1)
= (−1)[x] F−[x]−1 g(x − [x]) − F−[x] g(x − [x] − 1)
= f (x),
which completes our proof.
3. Hyers-Ulam stability of Fibonacci equation
As already stated, α denotes the positive root of the quadratic equation x2 − x − 1 = 0 and β is its negative root. We can prove the
Hyers-Ulam stability of the Fibonacci functional equation (1.1) as we
see in the following theorem.
Theorem 3.1. Let (X, k · k) be a real Banach space. If a function
f : R → X satisfies the inequality,
kf (x) − f (x − 1) − f (x − 2)k ≤ ε,
(3.1)
for all x ∈ R and for some ε > 0, then there exists a Fibonacci function
G : R → X such that
2
(3.2)
kf (x) − G(x)k ≤ 1 + √ ε,
5
for all x ∈ R.
Proof. Analogous to the first equation of (2.2), we get from (3.1):
kf (x) − αf (x − 1) − β[f (x − 1) − αf (x − 2)]k ≤ ε,
222
Jung
for each x ∈ R. If we replace x by x − k in the last inequality, then we
have,
kf (x − k) − αf (x − k − 1) − β[f (x − k − 1) − αf (x − k − 2)]k ≤ ε
and furthermore,
kβ k [f (x − k) − αf (x − k − 1)]
− β k+1 [f (x − k − 1) − αf (x − k − 2)]k
(3.3)
≤ |β|k ε,
for all x ∈ R and k ∈ Z. By (3.3), we obviously have,
kf (x) − αf (x − 1) − β n [f (x − n) − αf (x − n − 1)]k
≤
n−1
X
kβ k [f (x − k) − αf (x − k − 1)]
k=0
− β k+1 [f (x − k − 1) − αf (x − k − 2)]k
(3.4)
≤
n−1
X
|β|k ε,
k=0
for x ∈ R and n ∈ N.
For any x ∈ R, (3.3) implies that {β n [f (x − n) − αf (x − n − 1)]}
is a Cauchy sequence (note that |β| < 1). Therefore, we can define a
function G1 : R → X by
G1 (x) = lim β n [f (x − n) − αf (x − n − 1)],
n→∞
since X is complete. In view of the above definition of G1 , we obtain:
G1 (x − 1) + G1 (x − 2)
= β −1 lim β n+1 [f (x − (n + 1)) − αf (x − (n + 1) − 1)]
n→∞
−2
+β
lim β n+2 [f (x −
n→∞
β −1 G1 (x) + β −2 G1 (x)
(n + 2)) − αf (x − (n + 2) − 1)]
=
= G1 (x),
for all x ∈ R. Hence, G1 is a Fibonacci function. If n goes to infinity,
then (3.4) yields:
√
3+ 5
ε,
(3.5)
kf (x) − αf (x − 1) − G1 (x)k ≤
2
for every x ∈ R.
Hyers-Ulam stability of Fibonacci functional equation
223
On the other hand, it also follows from (3.1) that
kf (x) − βf (x − 1) − α[f (x − 1) − βf (x − 2)]k ≤ ε
(see the second equation in (2.2)). Analogous to (3.3), replacing x by
x + k in the above inequality, we have,
kα−k [f (x + k) − βf (x + k − 1)]
− α−k+1 [f (x + k − 1) − βf (x + k − 2)]k
(3.6)
≤ α−k ε,
for all x ∈ R and k ∈ Z. By using (3.6), we further obtain:
kα−n [f (x + n) − βf (x + n − 1)] − [f (x) − βf (x − 1)]k
n
X
≤
kα−k [f (x + k) − βf (x + k − 1)]
k=1
− α−k+1 [f (x + k − 1) − βf (x + k − 2)]k
(3.7)
≤
n
X
α−k ε,
k=1
for x ∈ R and n ∈ N.
On account of (3.6), we see that {α−n [f (x + n) − βf (x + n − 1)]} is a
Cauchy sequence, for any fixed x ∈ R. Hence, we can define a function
G2 : R → X by
G2 (x) = lim α−n [f (x + n) − βf (x + n − 1)].
n→∞
Using the above definition of G2 , we get:
G2 (x − 1) + G2 (x − 2)
= α−1 lim α−(n−1) [f (x + n − 1) − βf (x + (n − 1) − 1)]
n→∞
−2
+α
lim α−(n−2) [f (x
n→∞
α−1 G2 (x) + α−2 G2 (x)
+ n − 2) − βf (x + (n − 2) − 1)]
=
= G2 (x),
for any x ∈ R. So, G2 is also a Fibonacci function. If we let n go to
infinity, then it follows from (3.7) that
√
5+1
ε,
(3.8)
kG2 (x) − f (x) + βf (x − 1)k ≤
2
for x ∈ R.
224
Jung
By (3.5) and (3.8), we have,
β
α
f (x) −
G
(x)
−
G
(x)
1
2
β−α
β−α
1
k(β − α)f (x) − [βG1 (x) − αG2 (x)]k
=
|β − α|
1
≤
kβf (x) − αβf (x − 1) − βG1 (x)k
α−β
1
+
kαG2 (x) − αf (x) + αβf (x − 1)k
α−β
2
√
ε,
≤
1+
5
for all x ∈ R. We now set:
G(x) =
β
α
G1 (x) −
G2 (x),
β−α
β−α
and it is not difficult to show that G is a Fibonacci function.
Before we prove the uniqueness of the Fibonacci function G of Theorem 3.1, we show the following result.
Lemma 3.2. Let (X, k · k) be a real normed space and let u, v ∈ X be
given. If kFn+1 u + Fn vk ≤ C, for all n ∈ N and for some C ≥ 0, then
αu + v = 0.
Proof. We have,
Fn kαu + vk = kFn+1 u + Fn v − Fn+1 u + αFn uk
≤ kFn+1 u + Fn vk + |Fn+1 − αFn | kuk
n+1
α
− β n+1
αn − β n ≤ C +
−α
kuk
α−β
α−β = C + |β|n kuk,
for all n ∈ N. Since |β| < 1 and Fn → ∞ as n → ∞, we obtain
αu + v = 0.
In the following theorem, we prove the uniqueness of the Fibonacci
function G of Theorem 3.1.
Hyers-Ulam stability of Fibonacci functional equation
225
Theorem 3.3. The Fibonacci function G : R → X of Theorem 3.1 is
unique.
Proof. For i ∈ {1, 2}, let Gi : R → X be a Fibonacci function satisfying
2
(3.9)
kf (x) − Gi (x)k ≤ 1 + √ ε,
5
for all x ∈ R. Due to Theorem 2.1, there exist functions gi : [−1, 1) → X
(i ∈ {1, 2}) such that
(3.10)
(
F[x]+1 gi (x − [x]) + F[x] gi (x − [x] − 1)
(x ≥ 0),
Gi (x) =
(−1)[x] F−[x]−1 gi (x − [x]) − F−[x] gi (x − [x] − 1) (x < 0),
for i ∈ {1, 2}.
Fix a t with 0 ≤ t < 1. It then follows from (3.9) that
kG1 (n + t) − G2 (n + t)k
≤ kG1 (n + t) − f (n + t)k + kf (n + t) − G2 (n + t)k
2
≤ 2 1 + √ ε,
5
for any n ∈ Z. Furthermore, by (3.10) and the last inequality, we obtain:
kFn+1 [g1 (t) − g2 (t)] + Fn [g1 (t − 1) − g2 (t − 1)]k
= kG1 (n + t) − G2 (n + t)k
2
≤ 2 1+ √ ε
5
and
kFn−1 [g1 (t) − g2 (t)] − Fn [g1 (t − 1) − g2 (t − 1)]k
= kG1 (−n + t) − G2 (−n + t)k
2
≤ 2 1 + √ ε,
5
for each n ∈ N.
According to Lemma 3.2, we have,
(
α[g1 (t) − g2 (t)] + [g1 (t − 1) − g2 (t − 1)] = 0,
−α[g1 (t − 1) − g2 (t − 1)] + [g1 (t) − g2 (t)] = 0
226
Jung
or
α
1
!
g1 (t) − g2 (t)
!
0
=
1 −α
g1 (t − 1) − g2 (t − 1)
!
.
0
Because −α2 − 1 6= 0, we conclude:
g1 (t) − g2 (t) = g1 (t − 1) − g2 (t − 1) = 0.
Since 0 ≤ t < 1 is arbitrary, we have g1 (t) = g2 (t), for any −1 ≤ t < 1;
i.e., it follows from (3.10) that G1 (x) = G2 (x), for all x ∈ R.
Acknowledgments
The author is grateful to Professor Changsun Choi who proved the
uniqueness of the Fibonacci function (Lemma 3.2 and Theorem 3.3).
This work was supported by National Research Foundation of Korea
Grant funded by the Korean Government (No. 2009-0071206).
References
[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math.
Soc. Japan 2 (1950) 64–66.
[2] J. Baker, J. Lawrence and F. Zorzitto, The stability of the equation f (x + y) =
f (x)f (y), Proc. Amer. Math. Soc. 74 (1979) 242–246.
[3] S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type,
Hadronic Press, Palm Harbor, 2003.
[4] G. L. Forti, Hyers-Ulam stability of functional equations in several variables,
Aequationes Math. 50 (1995) 143–190.
[5] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14
(1991), 431–434.
[6] P. Gǎvrutǎ, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431–436.
[7] R. Ger and P. Šemrl, The stability of the exponential equation, Proc. Amer.
Math. Soc. 124 (1996) 779–787.
[8] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad.
Sci. USA 27 (1941) 222–224.
[9] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in
Several Variables, Birkhäuser, Boston, 1998.
[10] D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes
Math. 44 (1992) 125–153.
[11] S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations, Dynamic Sys.
Appl. 6 (1997) 541–566.
[12] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
Hyers-Ulam stability of Fibonacci functional equation
227
[13] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons,
New York, 2001.
[14] L. Maligranda, A result of Tosio Aoki about a generalization of Hyers-Ulam
stability of additive functions – a question of priority, Aequationes Math. 75
(2008) 289–296.
[15] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc.
Amer. Math. Soc. 72 (1978) 297–300.
[16] Th. M. Rassias, On the stability of functional equations and a problem of Ulam,
Acta Appl. Math. 62 (2000) 23–130.
[17] Th. M. Rassias, On the stability of functional equations in Banach spaces, J.
Math. Anal. Appl. 251 (2000) 264–284.
[18] L. Székelyhidi, On a theorem of Baker, Lawrence and Zorzitto, Proc. Amer.
Math. Soc. 84 (1982) 95-96.
[19] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ., New
York, 1960.
Soon-Mo Jung
Mathematics Section, College of Science and Technology, Hongik University,
339-701 Jochiwon, Republic of Korea.
Email: [email protected]